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<h2>曲线/曲面积分</h2>

<table>
  <tr>
    <td>第一类曲线积分</td>
    <td>`int_L f(x,y,z) "d"s`</td>
    <td>第二类曲线积分</td>
    <td>`int_L bm F(x,y,z) * "d"bm r`</td>
  </tr>
  <tr>
    <td>第一类曲面积分</td>
    <td>`int_Sigma f(x,y,z) "d"sigma`</td>
    <td>第二类曲面积分</td>
    <td>`int_Sigma bm F(x,y,z) * "d"bm S`</td>
  </tr>
</table>

<p>形式上有</p>

<table>
  <tr>
    <td>曲线切向微元</td>
    <td>`"d"bm r = (dx, dy, dz)`</td>
    <td>曲线弧长微元</td>
    <td>`"d"s = |"d"bm r|`</td>
  </tr>
  <tr>
    <td>曲面法向微元</td>
    <td>`"d"bm S = (dy dz, dz dx, dx dy)`</td>
    <td>曲面面积微元</td>
    <td>`"d"sigma = |"d"bm S|`</td>
  </tr>
</table>

<p>
  实际计算时, 为曲线 `L` (曲面 `Sigma`) 选取合适的参数, 就有
  <span class="formula">
    `"d"bm r = bm v dt`, `quad "d"s = |bm v| dt`;<br/>
    `"d"bm S = bm n "d"u"d"v`, `quad "d"sigma = |bm n|"d"u"d"v`.
  </span>
  其中
  <span class="formula">
    切向量 `bm v = ("d"bm r)/dt` `= (dx/dt, dy/dt, dz/dt)`, `quad`
    法向量 `bm n = bm r_u xx bm r_v` `= |bm i, bm j, bm k; x_u, y_u, z_u; x_v, y_v, z_v|`
  </span>
  依赖于参数选取.
</p>

<ol class="remark">
  <li>
  把曲面的单位法向量记为 `(cos alpha, cos beta, cos gamma)`, 则
  <span class="formula">
    `"d"bm S = (cos alpha, cos beta, cos gamma) "d"sigma`.
  </span>
  </li>
  <li>对于平面曲线, 其法向微元定义为 `"d"bm n = (dy, -dx)`,
    从而 `("d"bm r, "d"bm n)` 成左手系, 即 `|dx, dy; dy, -dx|
    = -"d"s^2 lt 0`.
  </li>
</ol>

<p class="example">
  求 `int_L x y "d"s`, 其中 `L: {x+y+z = 0; x^2+y^2+z^2=1:}`
</p>

<p class="solution">
  利用积分区域的对称性,
  <span class="formula">
    `int_L x y "d"s`
    `= 1/3 int_L (x y+y z+z x) "d"s`
    `= 1/6 int_L [(x+y+z)^2 - (x^2+y^2+z^2)] "d"s`
    `= -1/6 int_L "d"s`
    `= -pi/3`.
  </span>
</p>

<h2>微分算子</h2>

<h3>Nabla 算子</h3>

<span class="formula">
  `grad := (del/(del x_1), cdots, del/(del x_m))`
</span>

<p class="definition">
  标量场 (数量场) `f` 具有梯度 `"grad" f = grad f`.
  梯度是矢量, 其与坐标选取无关.
  梯度是导数的推广, 它简单地将求导运算推广到各分量,
  其加减乘除及函数复合的运算和导数完全一致.
</p>

<p>方向导数定义为函数值沿某个方向的变化率:
  <span class="formula">
    `("d"f)/("d"bm n) = lim_(t to 0) (f(bm x_0 + t bm n) - f(bm x_0))//t`
  </span>
  可以证明 `("d"f)/("d"bm n) = bm n * grad f`,
  因此梯度方向是该点处方向导数取得最大的方向, 大小等于这个最大的方向导数.
</p>

<table>
  <tr>
    <td>`grad c = 0`</td>
    <td>`grad (cf) = c grad f`</td>
  </tr>
  <tr>
    <td>`grad (f +- g) = grad f +- grad g`</td>
    <td>`grad (fg) = g grad f + f grad g`</td>
  </tr>
  <tr>
    <td>`grad (f//g) = (g grad f - f grad g) // g^2`</td>
    <td>`grad f(g) = ("d"f)/("d"g) grad g`</td>
  </tr>
  <tr>
    <td>`grad f(g_1, cdots, g_n) = sum (del f)/(del g_i) grad g_i`</td>
  </tr>
</table>

<p class="definition">
  矢量场 (向量场) `bm F` 具有散度
  <span class="formula">
    `"div" bm F = grad * bm F`
    `= (del F_1)/(del x_1) + cdots + (del F_n)/(del x_n)`.
  </span>
  和旋度 (三维空间):
  <span class="formula">
    `"rot" bm F = grad xx bm F`
    `= |bm i, bm j, bm k; del/(del x), del/(del y), del/(del z); F_1, F_2, F_3|`.
  </span>
  散度是标量而旋度是矢量.
</p>

<p>
  散度定义为封闭曲面包围区域趋于无穷小时, 单位体积的通量之极限,
  规定向外为正方向:
  <span class="formula">
    `"div" bm F = lim_("diam"V to 0) (iint_(del V) bm F * "d"bm S)/V`.
  </span>
  散度的符号指出矢量场在一点处有正源/负源, 散度的大小指出源的强度.
</p>

<p>
  方向旋量定义为给定法向量的平面上单位面积的环量之极限,
  规定右手螺旋为正方向:
  <span class="formula">
    `"rot"_(bm v) bm F = lim_("diam"S to 0) (int_(del S) bm F * "d"bm r)/S`.
  </span>
  `bm F` 与切向 `"d"bm r` 方向越近似, 环量就越是取正值.
  因此环量反映了矢量场沿曲线是否有旋涡, 以及旋涡的强度.
  方向旋量反映了矢量场在一点处沿给定方向的旋转情况, 显然只要 `bm v`
  取反方向, 方向旋量的符号就取反.
  由 Stokes 公式可以证明 `"rot"_(bm v) bm F = bm v * "rot" bm F`.
  旋度的方向是该点处方向旋量取得最大的方向, 大小等于这个最大的方向旋量.
</p>

<table>
  <tr>
    <td>`grad * bm c = 0`</td>
    <td>`grad xx bm c = bm 0`</td>
  </tr>
  <tr>
    <td>`grad * (c bm F) = c (grad * bm F)`</td>
    <td>`grad xx (c bm F) = c (grad xx bm F)`</td>
  </tr>
  <tr>
    <td>`grad * (bm F +- bm G) = grad * bm F +- grad * bm G`</td>
    <td>`grad xx (bm F +- bm G) = grad xx bm F +- grad xx bm G`</td>
  </tr>
  <tr>
    <td>`grad * (f bm F) = grad f * bm F + f (grad * bm F)`</td>
    <td>`grad xx (f bm F) = grad f xx bm F + f(grad xx bm F)`</td>
  </tr>
</table>

<p class="proposition">
  设 `f`, `bm F` 二阶连续可微, 因此二阶混合偏导相等.
  梯度场的散度是 laplace 算子 (见下节);
  梯度场的旋度为零向量, 旋度场的散度为 0.
  <span class="formula">
    `grad * (grad f) = (grad * grad) f := grad^2 f`,<br>
    `grad xx (grad f) = bm 0`,
    `quad grad * (grad xx bm F) = 0`.
  </tr>
</p>

<p class="definition">
  `bm G * grad` 算子可以用于标量或矢量. 输入标量, 输出标量; 输入矢量, 输出矢量:
  <span class="formula">
    `(bm G * grad) f`
    `= G_1 (del f)/(del x_1) + cdots + G_n (del f)/(del x_n)`,<br>
    `(bm G * grad) bm F`
    `= ((bm G * grad) F_1, cdots, (bm G * grad) F_n)`.
  </span>
</p>

<p>`D` 算子作用于矢量, 得到它的导矩阵:
  <span class="formula">
    `D bm F`
    `= [grad F_1; vdots; grad F_n]`
    `= [
      (del F_1)/(del x_1), cdots, (del F_1)/(del x_n);
      vdots, , vdots;
      (del F_n)/(del x_1), cdots, (del F_n)/(del x_n);
    ]`
  </span>
  作用于标量时, 就是 `D f = grad f`.
</p>

<ol class="proposition">
  <b>nabla 算子的综合公式</b>
  以下公式都涉及旋度, 因此只限于三维空间:
  <li>`grad(bm F * bm G)` `= bm F D bm G + bm G D bm F`
    `= bm F xx (grad xx bm G) + bm G xx (grad xx bm F) + (bm F * grad) bm G + (bm G * grad) bm F`;
  </li>
  <li>`grad * (bm F xx bm G) = bm G * grad xx bm F + bm F * grad xx bm G`;</li>
  <li>`grad xx (bm F xx bm G) = (grad * bm G) bm F - (grad * bm F) bm G + (bm G * grad) bm F - (bm F * grad) bm G`;</li>
  <li>`grad xx (grad xx bm F) = grad(grad * bm F) - (grad * grad)bm F`</li>
</ol>

<p class="proof">
  直接计算验证. 注意到各个分量的地位对称, 因此只验证 `x` 分量相等即可.
  以 1. 的证明为例, 左边的 `x` 分量等于
  <span class="formula">
    `F_1 (del G_1)/(del x) + F_2 (del G_2)/(del x) + F_3 (del G_3)/(del x)`
    `+ ** + ** + **`.
  </span>
  右边的 `x` 分量等于
  <span class="formula">
    `|F_2, F_3; (del G_1)/(del z) - (del G_3)/(del x), (del G_2)/(del x) - (del G_1)/(del y)|`
    `+ (F_1 del/(del x) + F_2 del/(del y) + F_3 del/(del z)) G_1`
    `+ | ** |`
    `+ **`.
  </span>
  其中 `**` 号省略的部分, 只需将字母 `F, G` 对换就可得到. 因此两式相等.
</p>

<h3>Laplace 算子</h3>

<span class="formula">
    `laplace := grad^2 = grad * grad = sum del^2/(del x_i^2)`
</span>

<table>
	<tr>
		<td>`laplace c = 0`</td>
	</tr>
	<tr>
		<td>`laplace (cf) = c laplace f`</td>
	</tr>
	<tr>
		<td>`laplace (f +- g) = laplace f +- laplace g`</td>
	</tr>
	<tr>
		<td>`laplace (fg) = f laplace g + g laplace f + 2 grad f * grad g`</td>
	</tr>
	<tr>
		<td>`laplace f(g) = grad ("d"f)/("d"g) * grad g + ("d"f)/("d"g) laplace g`</td>
	</tr>
	<tr>
		<td>`laplace f(g_1, cdots, g_n) = sum grad (del f)/(del g_i) * grad
		g_i + sum (del f)/(del g_i) laplace g_i`</td>
	</tr>
</table>

<p class="proof">
  如 `laplace(f g) = grad^2(f g)`
  `= grad * (f grad g + g grad f)`
  `= f grad^2 g + g grad^2 f + 2 grad f * grad g`.
</p>

<p>满足 `laplace f -= 0` 的函数称为<b>调和函数</b>.</p>

<h3>径向函数</h3>

<p>	设 `bm r = (x_1, cdots, x_m)`, `r = |bm r|`, 则
	<span class="formula">
		`grad r = (bm r)/r`, `quad |grad r| = 1`,
        `quad grad * bm r = m`, `quad laplace r = (m-1)/r`;<br/>
		`grad f(r) = ("d"f)/("d"r) grad r`,
		`quad laplace f(r) = ("d"^2 f)/("d"r^2) + (m-1)/r ("d"f)/("d"r)`;<br/>
	</span>
</p>

<p class="proof">
    `laplace r = grad * grad r`
    `= grad * (bm r)/r`
    `= 1/r grad * bm r + bm r * grad 1/r`
    `= m/r - 1/r^2 bm r * grad r`
    `= m/r - 1/r^3 bm r * bm r`
    `= (m-1)/r`.<br/>
    `laplace f(r) = grad * grad f(r)`
    `= grad ("d"f)/("d"r) * grad r + ("d"f)/("d"r) laplace r`
    `= ("d"^2 f)/("d"r^2) grad r * grad r + ("d"f)/("d"r) laplace r`
    `= ("d"^2 f)/("d"r^2) + (m-1)/r ("d"f)/("d"r)`.
</p>

<p> 特别当 `m = 3`, 在三维空间中有
    <span class="formula">
		`r laplace f = "d"^2/("d"r^2) (rf)`,
    </span>
    因此, `1//r` 是三维空间中的调和函数; 类似可证 `ln r`
    是二维空间的调和函数. 一般地, 微分方程
    <span class="formula">
        `f'' + (m-1)/r f' = 0`
    </span>
    的通解 `{ c_1 x^(2-m) + c_2, if m != 2;
    c_1 ln x + c_2, if m = 2:}` 给出 `m` 维空间的调和径向函数.
</p>

<h2>多元积分公式</h2>

<p>在二维空间中, 约定 `"d"V = dx dy`, 三维空间中则 `"d"V = dx dy dz`.</p>

<h3>Green, Gauss, Stokes</h3>

<p>	Leibniz:
	<span class="formula">
		`"d"/dt int_(alpha(t))^(beta(t)) f(x, t) dx`
		`= int_(alpha(t))^(beta(t)) del/(del t) f(x, t) dx`
		`+ f(beta(t), t) beta'(t)`
		`- f(alpha(t), t) alpha'(t).`
	</span>
	Green (记忆: `del y` 者符号相反):
	<span class="formula">
		`iint_V |del/(del x), del/(del y); P, Q| "d"V
		= oint_(del V) P dx + Q dy`
	</span>
	Gauss:
	<span class="formula">
		`iiint_V grad * bm F "d"V`
		`= oiint_(del V) bm F * "d" bm S`
	</span>
	Stokes:
	<span class="formula">
		`iint_Sigma grad xx bm F * "d" bm S
        = oint_(del Sigma) bm F * "d" bm r`,
	</span>
</p>

<p class="remark">
    用混合积的定义, Stokes 公式可以写为
    <span class="formula">
		`iint_S |
			del/(del x), del/(del y), del/(del z);
			P, Q, R;
			dy dz, dz dx, dx dy;
		|
		= oint_(del S) P dx + Q dy + R dz`,
    </span>
    取 `dz = 0` 就得到 Green 公式.
</p>

<p class="example">
  [来自 群Scalar] 求 `(t - t^3, 1 - t^4)`, `t in [-1, 1]` 围成图形的面积.
</p>

<p class="solution">
  由 Green 公式 (注意 `t` 从 `-1` 到 `1` 曲线为顺时针)
  <span class="formula">
    `iint_V dx dy`
    `= 1/2 oint_(del V) x dy - y dx`
    `= 1/2 int_1^(-1) [(t - t^3)(-4 t^3) - (1-t^4)(1-3t^2)] dt`
    `= 16/35`.
  </span>
</p>

<h3>Green 三大公式</h3>

<p> 回忆方向导数的定义: `(del f)/(del bm n) = grad f * bm n`,
    在 Gauss 公式中取 `bm F = grad f`, 则
    <span class="formula">
        `bm F * "d"bm S`
        `= grad f * bm n |"d"bm S|`
        `= (del f)/(del bm n) "d"sigma`,
    </span>
    得到有用的方向导数形式:
    <span class="formula">
		`iiint_V Delta f "d"V
		= oiint_(del V) (del f)/(del bm n) "d" sigma`,
		`quad bm n` 是单位外法向量.
    </span>
    在 Green 公式中取 `(Q, -P) = grad f`, 也得到方向导数形式:
    <span class="formula">
		`iint_V Delta f "d"V = oint_(del V) (del f)/(del bm n) "d"s`,
        `quad bm n` 是单位外法向量 `(dy/("d"s), -dx/("d"s))`.
    </span>
    这个公式实际是下面 Green 第一公式的特殊情形.
</p>

<ol class="theorem">
    <b>Green 第一公式</b>
    类比于分部积分公式将导数从一个因子转移到另一个因子上,
    此公式将 nabla 算子从 `g` 转移到 `f` 上:
    <li>(2d) `iint_V grad f * grad g "d"V
		= int_(del V) g{::} (del f)/(del bm n) "d"s
        - iint_V g laplace f "d"V`;
    </li>
    <li>(3d) `iiint_V grad f * grad g "d"V
		= iint_(del V) g{::} (del f)/(del bm n) "d"sigma
        - iiint_V g laplace f "d"V`.
    </li>
</ol>

<p class="proof">
    只证 2d 情形 (3d 情形其实更简单).
    利用方向导数的定义和 `bm n = (dy/("d"s), -dx/("d"s))`,
    <span class="formula">
        `int_(del V) g{::} (del f)/(del bm n) "d"s`
        `= int_(del V) g grad f * bm n "d"s`
        `= int_(del V) g ((del f)/(del x) dy - (del f)/(del y) dx)`
        `= iint_V [del/(del x)(g{::}(del f)/(del x))
        + del/(del y)(g{::}(del f)/(del y))] "d"V`
        `= iint_V grad f * grad g "d"V + iint_V g laplace f "d"V`.
    </span>
</p>

<ol class="theorem">
    <b>Green 第二公式</b>
    <li>(2d) `iint_V |laplace f, laplace g; f, g| "d"V
        = int_(del V) |(del f)/(del bm n), (del g)/(del bm n);
        f, g| "d"s`;
    </li>
    <li>(3d) `iiint_V |laplace f, laplace g; f, g| "d"V
        = iint_(del V) |(del f)/(del bm n), (del g)/(del bm n);
        f, g| "d"sigma`.
    </li>
</ol>

<p class="proof">
    分别对函数 `f, g` 应用 Green 第一公式即可.
    `grad f * grad g` 的积分刚好抵消.
</p>

<ol class="theorem">
    <b>Green 第三公式</b>
    设 `u` 为调和函数, `r` 是点 `(x,y)` 或 `(x,y,z)` 到 `del V`
    上积分变动点的距离.
    <li>(2d) `u(x,y) = 1/(2pi) int_(del V) (u (del ln r)/(del bm n)
        - ln r (del u)/(del bm n)) "d"s`;
    </li>
    <li>(3d) `u(x,y,z) = 1/(4pi) iint_(del V) (1/r (del u)/(del bm n)
        - u (del(1//r))/(del bm n)) "d"sigma`.
    </li>
</ol>

<p class="proof">
    只证 2d 情形.
    取 `C` 是以 `(x,y)` 为心, `rho` 为半径的圆周.
    对函数 `u` 和 `ln r`, 在 `C` 和 `del V` 所夹的区域上应用 Green
    第二公式, 注意 `ln r` 是二维空间的调和函数, 有
    <span class="formula">
        `int_(del V) - int_C = 0`,
    </span>
    其中被积函数是 `|(del ln r)/(del bm n), (del u)/(del bm n);
    ln r, u|`.
    于是
    <span class="formula">
        `int_(del V)`
        `= int_C (u (del ln r)/(del bm n) - ln r (del u)/(del bm n)) "d"s`
        `= int_C (u (del ln r)/(del r) - ln rho (del u)/(del bm n)) "d"s`
        `= 1/rho int_C u "d"s`.
    </span>
    令 `rho to 0`, 则 `u` 在 `C` 上的平均值
    `1/(2pi rho) int_C u "d"s` 趋于 `u(x,y)`,
    从而 `u(x,y) = 1/(2pi) int_(del V)`.
</p>

<p class="example">
	<b>Dirichlet 原理</b> 在区域边界上取给定值的连续可微函数,
	其 Dirichlet 积分 (函数的梯度的模的平方在区域上的积分)
	取最小值当且仅当该函数为调和函数.
</p>

<p class="proof">
	设 `laplace f = 0`, 且 `f|{::}_(del V) = g|_(del V)`.
	在 Green 第一公式中令 `g = f`, 利用边界条件得
	<span class="formula">
		`iint_V |grad f|^2 "d"V`
		`= int_(del V) f{::} (del f)/(del bm n) "d"s`
		`= int_(del V) g{::} (del f)/(del bm n) "d"s`
		`= iint_V grad f * grad g "d"V`.
	</span>
	从而
	<span class="formula">
		`iint_V |grad g|^2 "d"V - iint_V |grad f|^2 "d"V`
		`= iint_V |grad g|^2 "d"V + iint_V |grad f|^2 "d"V
        - 2 iint_V grad f * grad g "d"V`
		`= iint_V |grad g - grad f|^2 "d"V ge 0`.
	</span>
</p>

<h2>微分形式与外微分运算</h2>

[来自 <a href="https://zhuanlan.zhihu.com/p/392443514">茶凉凉凉凉</a>]

<ol class="definition">
  <b>外积</b>
  观察多元积分换元公式
  <span class="formula">
    `"d"u"d"v = (del(u, v))/(del(x, y)) dx dy`,
  </span>
  如果将 `x, y` 的次序对调, 得到
  <span class="formula">
    `dy dx = (del(y, x))/(del(x, y)) dx dy`
    `= |y_x, y_y; x_x, x_y| dx dy`
    `= |0, 1; 1, 0| dx dy`
    `= -dx dy`.
  </span>
  这个性质类似于向量的外积, 我们也把它称为外积, 写作 `dx ^^ dy`.
  外积运算满足
  <li>线性性. `(P dx + Q dy) ^^ "d"u = P dx ^^ "d"u + Q dy ^^ "d"u`;</li>
  <li>反对称性. `dx ^^ dy = - dy ^^ dx`.</li>
  由反对称性立即得到 `dx ^^ dx = 0`.<br>
  一般地, 在 `n` 维空间中, 规定每交换一对变量, 外积就改变一次符号:
  <span class="formula">
    `dx_1 ^^ cdots ^^ color(red)(dx_i) ^^ cdots ^^ color(blue)(dx_j) ^^ cdots ^^ dx_n`
    `= -dx_1 ^^ cdots ^^ color(blue)(dx_j) ^^ cdots ^^ color(red)(dx_i) ^^ cdots ^^ dx_n`
  </span>
</ol>

<ol class="definition">
  <li><b>零阶微分形式</b> 就是普通的多元函数 `f`.</li>
  <li><b>一阶微分形式</b> 是 `dx, dy, dz` 以及它们的线性组合 (系数是普通函数, 如 `P dx + Q dy`).</li>
  <li><b>二阶微分形式</b> 是 `dx ^^ dy`, `dy ^^ dz`, `dz ^^ dx`
  以及它们的线性组合.</li>
  <li><b>三阶微分形式</b> 是 `dx ^^ dy ^^ dz` 以及它的线性组合 (其实在三维空间中只有一种组合).</li>
  由 `dx ^^ dx = 0` 知道, 在 `n` 维空间中, 最高只有 `n` 阶微分形式.
</ol>

<ol class="definition">
  设 `omega` 是微分形式, 定义<b>外微分</b>运算 `"d"omega` 如下:
  <li>零阶微分形式 `f` 的外微分定义为它的全微分:
    <span class="formula">
      `"d"f = grad f * "d"bm r`
      `= sum (del f)/(del x_i) dx_i`.
    </span>
  </li>
  <li>若 `omega` 形如 `omega = dx_1 cdots dx_n`, 则单项式 `f omega`
    的外微分定义为
    <span class="formula">
      `"d"(f omega) = "d"f ^^ "d"omega`.
    </span>
    利用外积运算的线性性, 可将 `"d"f` 展开各项与 `"d"omega` 相乘,
    再合并同类项.
  </li>
  <li>最后, 规定外微分具有线性性
    <span class="formula">
      `"d"(omega_1 + omega_2) = "d"omega_1 + "d"omega_2`,
    </span>
    欲求整个微分形式的外微分, 只需将各个单项式的外微分相加.
  </li>
</ol>

<ol class="example">
  <b>外微分下的三大公式</b>
  <li>Green: `"d"(P dx + Q dy)`
    `= ((del Q)/(del x) - (del P)/(del y)) dx ^^ dy`.
  </li>
  <li>Stokes: `"d"(P dx+Q dy+R dz)`
    `= ((del R)/(del y)-(del Q)/(del z)) dy ^^ dz`
    `+ ((del P)/(del z)-(del R)/(del x)) dz ^^ dx`
    `+ ((del Q)/(del x)-(del P)/(del y)) dx ^^ dy`.
  </li>
  <li>Gauss: `"d"(P dy^^dz + Q dz^^dx + R dx^^dy)`
    `= ((del P)/(del x) +(del Q)/(del y) + (del R)/(del z)) dx ^^ dy ^^ dz`.</li>
  至此三大公式, 包括 Newton-Leibniz 公式都统一写成:
  <span class="formula">
    `int_(del D) omega = int_D "d" omega`.
  </span>
</ol>

<ol class="proof">
  <li>左边等于
    <span class="formula">
      `"d"P ^^ dx + "d"Q ^^ dy`
      `= ((del P)/(del x) dx + (del P)/(del y) dy) ^^ dx`
      `+ ((del Q)/(del x) dx + (del Q)/(del y) dy) ^^ dy`
    </span>
    等于右边.
  </li>
  <li>和 Green 公式完全类似.</li>
  <li>第一项
    <span class="formula">
      `"d"P ^^ dy ^^ dz`
      `= ((del P)/(del x)dx+(del P)/(del y)dy+(del P)/(del z)dz) ^^ dy ^^
      dz`
      `= (del P)/(del x) dx ^^ dy ^^ dz`.
    </span>
    其余两项是类似的 (注意 `dx ^^ dy ^^ dz = dy ^^ dz ^^ dx = dz ^^ dx ^^
    dy`).
  </li>
</ol>

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